How to get Better and Better Rational Approximations to Pi Without Cheating

In fond memory of Albert Einstein (March 14, 1879-∞).
[Recall that Pi is one of the three constants featuring in his famous Field Equation]

Dowload the E-book: .txt   (4.35 MB)

(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger)

Generated March 14, 2014

Unfortunately, Pi is not a ratio of integers, but one can try to get close. One way is to use any of the fast algorithms (cf the Borweins' classic, Pi and the AGM), or the amazingly fast-convergent series, e.g. Jesús Guillera's amazing series to compute decimal approximations, and then have the `best' rational approximations where the denominators are powers of 10. An even better, but still cheating, way, is to convert the floating-point approximation into a continued fraction, and get convergents. But we can't prove that this goes for ever.

To generate this gripping E-book, the author executed the input file, that called the maple code, written by Doron Zeilberger, that, in turn, calls Zeilberger's Maple packages EKHAD (that contains the Almkvist-Zeilberger algorithm), and AsyRec, to genetate the E-book.

Alas, this sequence is not good enough to prove irrationality, since the denominators grow too fast, since the δ such that

|π-an/bn| ≤ C/bnδ

is not larger than 1, failing to give an Apéry-style irrationality proof (and measure), yet it is still pas mal, almost one half!

Happy Pi Day! Happy birthday, Albert!